arXiv2026
We derive equations of motion for capital deepening in a competitive economy directly from accounting identities, without assuming a production function. A profit imperative $η^* \equiv (w/κ+ 1/τ)/(1-f_p)$ sets the minimum viable capital productivity, where $η= Y/K$ [yr$^{-1}$] is capital productivity, $κ= K/L$ is capital per worker, $w$ is the wage rate, $τ$ is the capital lifetime, and $f_p$ is the production tax share. Four coupled relaxation equations govern $κ$, $η$, the frontier productivity $η_{\rm new}$ of new investment, and the labor share $q \equiv w/y$, with the sandwich constraint $η^* \leq η_{\rm new} \leq η$ maintained as an exact invariant. The frontier equation separates two physically distinct channels: a structural cheapening channel ($μ$, always active, drives $η_{\rm new}$ downward) and a productivity channel ($ϕ$, historically zero). Calibration against BEA 2-digit NAICS sector data (1998--2023) confirms $ϕ= 0$ for all identifiable sectors over 25 years; the 75-year postwar record extends this finding across four capital lifetimes. A step $ϕ= 0.01$\,yr$^{-1}$ -- a 1\%/yr improvement in new-capital productivity, modest but historically unprecedented -- nearly doubles the aggregate growth rate within one capital lifetime, a falsifiable prediction with a precise observable signature: upward-curving $η(t)$ in BEA sector data. Firms near the zero-profit threshold have a cash martingale, predicting establishment exit rate $\sim t^{-1/2}$; convolved with the Zipf firm-size distribution~\cite{WP}, this yields firm exit rate $\sim t^{-1/2}\!\log t$ with apparent exponent $b = 0.295 \pm 0.03$, confirmed against BDS data with no free parameters.